2
$\begingroup$

enter image description here The definition of a straight line according to google.

I do not understand why I call these geometries "non-Euclidean". In my view, both hyperbolic and elliptical geometry are just a dimensional reference change of the plane, using the same elements described by Euclid. Both are described by curved planes, that is, analyzed three-dimensionaly. A straight line is no longer a straight line. Perhaps there is a lost axiom that has not been introduced to better define what a line is and not to confuse it with a curve. What I want to mean is that, all of these are the same elements but with another perspective. If we can define what a line really is, maybe we can debunk the axioms denying the parallels axiom. If anyone has understood my doubt, please tell me where I am wrong or if there is truth in my words. Thanks.

$\endgroup$
17
  • 4
    $\begingroup$ In euclidean geometry the parallel postulate holds, in non-euclidean ones it doesn't. I doubt "dimensional reference change" and/or "analyzed three dimensionality" mean anything... The reason non-euclidean geometries are non-euclidean is extremely concrete and simple. $\endgroup$
    – Mariano Suárez-Álvarez
    Apr 1, 2018 at 4:47
  • 1
    $\begingroup$ I suggest you read a modern exposition of euclidean geometry. Axioms do not define what a line is, for example —lines are undefined concepts, and that is in fact most of the point! $\endgroup$
    – Mariano Suárez-Álvarez
    Apr 1, 2018 at 4:49
  • 3
    $\begingroup$ If Lucky's comment is a correct interpretation of your motivation, it's worth noting that the hyperbolic plane cannot in fact be isometrically embedded into $\mathbb{R}^3$; this is due to Hilbert. $\endgroup$
    – Noah Schweber
    Apr 1, 2018 at 5:29
  • 1
    $\begingroup$ @spaceisdarkgreen, that is simply not true. There are models of euclidean geometry in which lines are not straight lines. A stupid way to get one is to find any bijection of the plane onto itself which does not preserve straight lines, and define a line to be the image of a usual line under that bijection. There are other, more clever models in which the plane is the inside of a disc, for example, and lines are parts of ellipses. $\endgroup$
    – Mariano Suárez-Álvarez
    Apr 1, 2018 at 5:43
  • 1
    $\begingroup$ @nicolas, you should really read up a bit about what euclidean and non-euclidean geometry actually is. $\endgroup$
    – Mariano Suárez-Álvarez
    Apr 1, 2018 at 5:46

1 Answer 1

Reset to default
2
$\begingroup$

The purpose of this comment is to illustrate that the concept of "being straight" is only a psychological one.

The figure below depicts the Klein model of the hyperbolic geometry within which the Euclidean plane can be modelled. Here the half ellipses centered at the center of the Klein circle tangent to the same are the Euclidean straights. And the hyperbolic straights "look straight".

The solid lines are Euclidean and the broken lines are hyperbolic. In this "Orthus" model the Euclidean and the hyperbolic geometry live together.

enter image description here

Native Hyperboleans will find the Euclidean straights to be funny and curved while the Euclideans will find the hyperbolic straights to be funny and curved.

The question: "Who is right?" is completely meaningless.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .